Question 2.5.3: An Example of Subsystems: Two Coupled Tanks Consider two bri...
An Example of Subsystems: Two Coupled Tanks
Consider two brine tanks each containing 500 L (liters) of brine connected as shown in Figure 2.5.1. At any time t, the first and the second tank contain x_1(t) and x_2(t) kg of salt, respectively. The brine concentration in each tank is kept uniform by continuous stirring. Brine containing r kg of salt per liter is entering the first tank at a rate of 15 L/min, and fresh water is entering the second tank at a rate of 5 L/min. The incoming brine density r(t) can be changed to regulate the process, so r(t) is an input variable.
Brine is pumped from the first tank to the second one at a rate of 60 L/min and from the second tank to the first one at a rate of 45 L/min. Brine is discharged from the second tank at a rate of 20 L/min.
a. Obtain the differential equations, in terms of x_1 and x_2, that describe the salt content in each tank as a function of time.
b. Obtain the transfer functions X_1(s)/R(s) and X_2(s)/R(s).
c. Suppose that r(t) = 0.2 kg/L. Determine the steady-state values of x_1 and x_2, and estimate how long it will take to reach steady state.